Variational regularisation is the primary method for solving inverse
problems, and recently there has been considerable work leveraging deeply
learned regularisation for enhanced performance. However, few results exist
addressing the convergence of such regularisation, particularly within the
context of critical points as opposed to global minima. In this paper, we
present a generalised formulation of convergent regularisation in terms of
critical points, and show that this is achieved by a class of weakly convex
regularisers. We prove convergence of the primal-dual hybrid gradient method
for the associated variational problem, and, given a Kurdyka-Lojasiewicz
condition, an $\mathcal{O}(\log{k}/k)$ ergodic convergence rate. Finally,
applying this theory to learned regularisation, we prove universal
approximation for input weakly convex neural networks (IWCNN), and show
empirically that IWCNNs can lead to improved performance of learned adversarial
regularisers for computed tomography (CT) reconstruction.

采用弱凸正则化方法，我们证明了原始 - 对偶混合梯度方法在关联的变分问题上收敛，并在满足 Kurdyka-Lojasiewicz 条件的情况下获得了 O (log (k)/k) 的遍历收敛速率。最后，我们将该理论应用于学习正则化，并证明了输入弱凸神经网络（IWCNN）的普适逼近性质，并实证了 IWCNN 在计算机断层扫描（CT）重建中学习对抗正则化方法的改进性能。

逆问题的弱凸正则化项：关键点的收敛性和原始对偶优化

Weakly Convex Regularisers for Inverse Problems: Convergence of Critical  Points and Primal-Dual Optimisation

In recent years the use of convolutional layers to encode an inductive bias
(translational equivariance) in neural networks has proven to be a very
fruitful idea. The successes of this approach have motivated a line of research
into incorporating other symmetries into deep learning methods, in the form of
group equivariant convolutional neural networks. Much of this work has been
focused on roto-translational symmetry of $\mathbf R^d$, but other examples are
the scaling symmetry of $\mathbf R^d$ and rotational symmetry of the sphere. In
this work, we demonstrate that group equivariant convolutional operations can
naturally be incorporated into learned reconstruction methods for inverse
problems that are motivated by the variational regularisation approach. Indeed,
if the regularisation functional is invariant under a group symmetry, the
corresponding proximal operator will satisfy an equivariance property with
respect to the same group symmetry. As a result of this observation, we design
learned iterative methods in which the proximal operators are modelled as group
equivariant convolutional neural networks. We use roto-translationally
equivariant operations in the proposed methodology and apply it to the problems
of low-dose computerised tomography reconstruction and subsampled magnetic
resonance imaging reconstruction. The proposed methodology is demonstrated to
improve the reconstruction quality of a learned reconstruction method with a
little extra computational cost at training time but without any extra cost at
test time.

本文介绍一种使用群等变卷积神经网络来解决逆问题的学习重建方法，通过在迭代方法中建立群等变卷积神经网络解决拉伸同变的问题，实现了低剂量计算机断层成像重建和子采样磁共振成像重建的质量提升。